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Theorem undi 3429
Description: Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undi  |-  ( A  u.  ( B  i^i  C ) )  =  ( ( A  u.  B
)  i^i  ( A  u.  C ) )

Proof of Theorem undi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3371 . . . 4  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
21orbi2i 505 . . 3  |-  ( ( x  e.  A  \/  x  e.  ( B  i^i  C ) )  <->  ( x  e.  A  \/  (
x  e.  B  /\  x  e.  C )
) )
3 ordi 834 . . 3  |-  ( ( x  e.  A  \/  ( x  e.  B  /\  x  e.  C
) )  <->  ( (
x  e.  A  \/  x  e.  B )  /\  ( x  e.  A  \/  x  e.  C
) ) )
4 elin 3371 . . . 4  |-  ( x  e.  ( ( A  u.  B )  i^i  ( A  u.  C
) )  <->  ( x  e.  ( A  u.  B
)  /\  x  e.  ( A  u.  C
) ) )
5 elun 3329 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
6 elun 3329 . . . . 5  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
75, 6anbi12i 678 . . . 4  |-  ( ( x  e.  ( A  u.  B )  /\  x  e.  ( A  u.  C ) )  <->  ( (
x  e.  A  \/  x  e.  B )  /\  ( x  e.  A  \/  x  e.  C
) ) )
84, 7bitr2i 241 . . 3  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ( x  e.  A  \/  x  e.  C ) )  <->  x  e.  ( ( A  u.  B )  i^i  ( A  u.  C )
) )
92, 3, 83bitri 262 . 2  |-  ( ( x  e.  A  \/  x  e.  ( B  i^i  C ) )  <->  x  e.  ( ( A  u.  B )  i^i  ( A  u.  C )
) )
109uneqri 3330 1  |-  ( A  u.  ( B  i^i  C ) )  =  ( ( A  u.  B
)  i^i  ( A  u.  C ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    u. cun 3163    i^i cin 3164
This theorem is referenced by:  undir  3431  dfif4  3589  dfif5  3590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172
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